Question: Find the least common multiple $(\text{LCM})$ of $14m^5-14m$ and $5m^5+5m^3$. You can give your answer in its factored form.
Answer: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $14m^5-14m$ can be factored as ${(2)(7)}{(m)(m^2+1)}{(m+1)(m-1)}$ by factoring out a $14m$ and using the difference of squares pattern twice. $5m^5+5m^3$ can be factored as ${(5)(m^2)}{(m)(m^2+1)}$ by factoring out a $5m^3$. We can see that: Both polynomials share the factors ${(m)(m^2+1)}$ Only the first polynomial has the factors ${(2)(7)(m+1)(m-1)}$ Only the second polynomial has the factors ${(5)(m^2)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(m)(m^2+1)}{(2)(7)(m+1)(m-1)}{(5)(m^2)}\\\\ &=70(m^3)(m+1)(m^2+1)(m-1)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $70(m^3)(m+1)(m^2+1)(m-1)$.